Between categories

To emphasize that one means a functor C→DC \to D as stated and not as a functor Cop→DC^{op} \to D one sometimes says covariant functor for non-contravariant, for emphasis.

Equivalently, a contravariant functor from CC to DD may be thought of as a functor from CC to DopD^op, but the version above generalises better to functors of many variables.

Also notice that while the objects of the functor category[Cop,D][C^{op}, D] are in canonical bijection with those in the functor category [C,Dop][C, D^{op}] (both are contravariant functors from CC to DD), the morphisms in the two functor categories are in general different, as

In a graded 22-category

The above definition of contravariant functors as covariant functors out of (or into) the opposite category is unsatisfactory if one wishes to consider natural transformations between functors of different variance. In particular, it seems to break the 22-categorical structure since with the above definition covariant and contravariant functors now live as objects in different categories [C,D][C,D] and [Cop,D][C^{op},D] inside the 22-category Cat of categories and covariant functors, hence whatever natural transformations between functors of different variance are, it is not 22-morphisms in Cat.

The correct thing to do is to define the ℤ/2\mathbb{Z}/2-graded $2$-category? of categories, functors, and natural transformations.

The objects of our 22-category will be categories.

The 11-morphisms of our 22-category will be all functors, covariant and contravariant alike. The grading (on the 11-categorical part at least) is then a functor to \mathbb Z/2 considered as one-object category, given by considering contravariant functors to be odd (1bmod2\mathbb Z1\bmod 2) and covariant functors to be even0bmod20\bmod 2. Composition of functors is then additive on the grading (as it should be).

The 22-morphisms F⇒αG:C→DF\stackrel{\alpha}{\Rightarrow}G\colon C\to D will be once again ob(C)\ob(C)-indexed families of morphisms FX→αXGXFX\stackrel{\alpha_X}{\to} GX, but the commutative diagram they will have to satisfy for each XYfX\stackrel{f} Y in CC depends on the combination of variances of FF and GG:

It is clear that (vertical) composition of 22-morphisms makes sense, and that furthermore natural transformations between functors of different variance should be graded odd, while natural transformation between functors of the same variance should be graded even.

It is less clear that (but true!) that horizontal composition and whiskering also make sense. We thus have almost a (strict) 22-category, if it were not for the fact that the actions of whiskering by an odd functor act contravariantly on vertical composition (so horizontal composition itself does not behave as usual), so perhaps this is a slightly more general notion of a 22-category.

What kind of notion of 22-category is this? I’ve been using the phrase ℤ/2\mathbb{Z}/2-graded 22-category, but this structure is not exactly a 22-category because of the allowed contravariance in the whiskering/horiztonal composition. –Vladimir_Sotirov

In this context, the structure of the ⋅op\cdot^{op} operation becomes interesting: it is an involutive 22-functor that preserves grading, but reverses 22-morphisms, and furthermore comes equipped with natural isomorphisms [−op,−]≅[−,−]op≅[−,−op][-^{op},-]\cong[-,-]^{op}\cong[-,-^{op}] (this doesn’t make complete sense as we have not discussed yet how our new 22-category is enriched in itself, so the meaning of [−,−]op[-,-]^{op} is not completely clear…).

Between 22-categories

As mentioned in opposite 2-category, a 22-category can have three different duals, depending on whether we formally flip only the 11-morphisms, only the 22-morphisms, or both. From the perspective in this article, however, it is better to say that 22-functors have three different kinds of contravariance (hence there are four kinds of 22-functors). Consequently, there should be two ℤ/2×ℤ/2\mathbb{Z}/2\times\mathbb{Z}/2-graded 22-categories (or maybe slightly more general structures since again how horizontal composition distributes over vertical composition will depend on the associated variances:

A ℤ/2×ℤ/2\mathbb{Z}/2\times\mathbb{Z}/2-graded 22-category of 22-categories, 22-functors of arbitrary variance, and lax natural transformations. Instead of giving each commutativity condition for the sixteen kinds of lax natural transformations, let us write down the one that a category of V-enriched categories comes equipped with: a lax natural transformation F⇒αG:F\stackrel{\alpha}\Rightarrow G\colonC→\toD, where FF is a 22-functor flipping 22-morphisms, and GG is 22-functor flipping 11-morphisms, consists of an ob(C)\ob(\mathbf{C})-indexed family of 11-morphisms FX→αXGXFX\stackrel{\alpha_X}{\to} GX in D, and for each two objects X,YX,Y of C, an ob[X,Y]\ob[X,Y]-indexed family of 22-morphisms αf\alpha_f, so that for every 22-morphism f⇒γgf\stackrel{\gamma}{\Rightarrow} g, we have the commutative diagram of 22-moprhisms in D:

where . is whiskering/horizontal composition. Furthermore, given composable 11-morphisms X→fY→hZX\stackrel{f}{\rightarrow}Y\stackrel{h}{\rightarrow} Z, the 22-moprhisms αX⇒αfGf∘αY∘Ff\alpha_X\stackrel{\alpha_f}{\Rightarrow}Gf\circ\alpha_Y\circ Ff and αY⇒αhGh∘αZ∘Fh\alpha_Y\stackrel{\alpha_h}{\Rightarrow}Gh\circ\alpha_Z\circ Fh are related via the formula αh∘f=(Gf.αh.Ff)∘αf\alpha_{h\circ f}=(Gf.\alpha_h.Ff)\circ\alpha_f, which says that the pasting diagram of 22-morphisms:

A ℤ/2×ℤ/2\mathbb{Z}/2\times\mathbb{Z}/2-graded 22-category of 22-categories, 22-functors of arbitrary variance, and oplax natural transformations.

Somewhat mysteriously, the category of V-enriched categories is a 22-category which comes with a unit enriched categoryℐ\mathcal{I} and either a lax natural transformation [ℐ,−]op⇒[[ℐ,−],V0][\mathcal{I},-]^{op}\Rightarrow[[\mathcal{I},-],V_0] (in the case of 𝒱\mathcal{V} a monoidal structure on VV), or a lax natural transformation [ℐ,−]op⇒[−,Ve][\mathcal{I},-]^{op}\Rightarrow[-,V^e] (in the case of 𝒱\mathcal{V} a closed structure on [ℐ,Ve]≅V0[\mathcal{I},V^e]\cong V_0).